Problem: The sum of two positive integers $a$ and $b$ is 1001.  What is the largest possible value of $\gcd(a,b)$?
Note that $\gcd(a,b)$ divides both $a$ and $b$, so $\gcd(a,b)$ must also divide $a + b = 1001$.  Clearly, $\gcd(a,b)$ cannot be equal to 1001 (because both $a$ and $b$ must be less than 1001).  The next-largest divisor of 1001 is 143.  If $a = 143$ and $b = 1001 - 143 = 858$, then $\gcd(a,b) = \gcd(143,858) = 143$.  Therefore, the largest possible value of $\gcd(a,b)$ is $\boxed{143}$.